A proof-theoretic treatment of λ-reduction with cut-elimination: λ-calculus as a logic programming language

2011 ◽  
Vol 76 (2) ◽  
pp. 673-699 ◽  
Author(s):  
Michael Gabbay
Keyword(s):  
Logic Programming ◽  
Programming Language ◽  
Functional Programming ◽  
Order Logic ◽  
First Order Logic ◽  
Cut Elimination ◽  
Logic Programming Language ◽  
First Order ◽  
Functional Programming Language ◽  
Sequent System

AbstractWe build on an existing a term-sequent logic for the λ-calculus. We formulate a general sequent system that fully integrates αβη-reductions between untyped λ-terms into first order logic.We prove a cut-elimination result and then offer an application of cut-elimination by giving a notion of uniform proof for λ-terms. We suggest how this allows us to view the calculus of untyped αβ-reductions as a logic programming language (as well as a functional programming language, as it is traditionally seen).

Visual And Diagrammatic Languages

2011 ◽  
pp. 22-50
Author(s):  
Bernd Meyer ◽  
Paolo Bottoni
Keyword(s):  
Logic Programming ◽  
Programming Language ◽  
Linear Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Visual Languages ◽  
New Approach ◽  
Logic Programming Language ◽  
First Order ◽  
Complex Forms

In this paper we investigate a new approach to formalizing interpretation of and reasoning with visual languages based on linear logic. We argue that an approach based on logic makes it possible to deal with different computational tasks in the usage of visual notations, from parsing and animation to reasoning about diagrams. However, classical first order logic, being monotonic, is not a suitable basis for such an approach. The paper therefore explores linear logic as an alternative. We demonstrate how parsing corresponds to linear proofs and prove the soundness and correctness of this mapping. As our mapping of grammars is into a subset of a linear logic programming language, we also demonstrate how multi-dimensional parsing can be understood as automated linear deduction. We proceed to discuss how the same framework can be used as the foundation of more complex forms of reasoning with and about diagrams.

Extended natural semantics

1993 ◽  
Vol 3 (2) ◽  
pp. 123-152 ◽  
Author(s):  
John Hannan
Keyword(s):  
Logic Programming ◽  
Programming Language ◽  
Functional Programming ◽  
Higher Order ◽  
Order Logic ◽  
Inference Rules ◽  
Abstract Syntax ◽  
Higher Order Logic ◽  
First Order ◽  
Definition Of

AbstractWe extend the definition of natural semantics to include simply typed λ-terms, instead of first-order terms, for representing programs, and to include inference rules for the introduction and discharge of hypotheses and eigenvariables. This extension, which we call extended natural semantics, affords a higher-level notion of abstract syntax for representing programs and suitable mechanisms for manipulating this syntax. We present several examples of semantic specifications for a simple functional programming language and demonstrate how we achieve simple and elegant manipulations of bound variables in functional programs. All the examples have been implemented and tested in λProlog, a higher-order logic programming language that supports all of the features of extended natural semantics.

Logics for the Semantic Web

2011 ◽  
pp. 24-43
Author(s):  
J. Bruijn
Keyword(s):  
Semantic Web ◽  
Logic Programming ◽  
Description Logic ◽  
Description Logics ◽  
Order Logic ◽  
First Order Logic ◽  
Horn Logic ◽  
First Order ◽  
Logical Language ◽  
Rule Languages

This chapter introduces a number of formal logical languages which form the backbone of the Semantic Web. They are used for the representation of both ontologies and rules. The basis for all languages presented in this chapter is the classical first-order logic. Description logics is a family of languages which represent subsets of first-order logic. Expressive description logic languages form the basis for popular ontology languages on the Semantic Web. Logic programming is based on a subset of first-order logic, namely Horn logic, but uses a slightly different semantics and can be extended with non-monotonic negation. Many Semantic Web reasoners are based on logic programming principles and rule languages for the Semantic Web based on logic programming are an ongoing discussion. Frame Logic allows object-oriented style (frame-based) modeling in a logical language. RuleML is an XML-based syntax consisting of different sublanguages for the exchange of specifications in different logical languages over the Web.

scholarly journals VERIFYING PROBABILISTIC PROGRAMS USING A HOARE LIKE LOGIC

2002 ◽  
Vol 13 (03) ◽  
pp. 315-340 ◽  
Author(s):  
J. I. DEN HARTOG ◽  
E. P. DE VINK
Keyword(s):  
Programming Language ◽  
Denotational Semantics ◽  
Order Logic ◽  
First Order Logic ◽  
Proof System ◽  
Probabilistic Choice ◽  
First Order ◽  
Probabilistic Programs

Probability, be it inherent or explicitly introduced, has become an important issue in the verification of programs. In this paper we study a formalism which allows reasoning about programs which can act probabilistically. To describe probabilistic programs, a basic programming language with an operator for probabilistic choice is introduced and a denotational semantics is given for this language. To specify propertics of probabilistic programs, standard first order logic predicates are insufficient, so a notion of probabilistic predicates is introduced. A Hoare-style proof system to check properties of probabilistic programs is given. The proof system for a sublanguage is shown to be sound and complete; the properties that can be derived are exactly the valid properties. Finally some typical examples illustrate the use of the probabilistic predicates and the proof system.

scholarly journals Compositional Neural Logic Programming

2021 ◽  
Author(s):  
Son N. Tran
Keyword(s):  
Neural Networks ◽  
Logic Programming ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Forward Chaining ◽  
Symbolic Reasoning ◽  
New Knowledge ◽  
Neural Logic

This paper introduces Compositional Neural Logic Programming (CNLP), a framework that integrates neural networks and logic programming for symbolic and sub-symbolic reasoning. We adopt the idea of compositional neural networks to represent first-order logic predicates and rules. A voting backward-forward chaining algorithm is proposed for inference with both symbolic and sub-symbolic variables in an argument-retrieval style. The framework is highly flexible in that it can be constructed incrementally with new knowledge, and it also supports batch reasoning in certain cases. In the experiments, we demonstrate the advantages of CNLP in discriminative tasks and generative tasks.

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